Theorem fnrnoprval 4114

Description: The range of an operation expressed as a collection of the operation's values.

Assertion

Ref	Expression
fnrnoprval	|- (F Fn (A X. B) -> ran F = {z | E.x e. A E.y e. B z = (xFy)})

Distinct variable groups:   x,y,z,A   x,B,y,z   x,F,y,z

Proof of Theorem fnrnoprval

Step	Hyp	Ref	Expression
1		fnrnfv 3835	. 2 |- (F Fn (A X. B) -> ran F = {z | E.w e. (A X. B)z = (F` w)})
2		fveq2 3800	. . . . . 6 |- (w = <.x, y>. -> (F` w) = (F` <.x, y>.))
3		df-opr 4041	. . . . . 6 |- (xFy) = (F` <.x, y>.)
4	2, 3	syl6eqr 1562	. . . . 5 |- (w = <.x, y>. -> (F` w) = (xFy))
5	4	eqeq2d 1523	. . . 4 |- (w = <.x, y>. -> (z = (F` w) <-> z = (xFy)))
6	5	rexxp 3276	. . 3 |- (E.w e. (A X. B)z = (F` w) <-> E.x e. A E.y e. B z = (xFy))
7	6	abbii 1612	. 2 |- {z | E.w e. (A X. B)z = (F` w)} = {z | E.x e. A E.y e. B z = (xFy)}
8	1, 7	syl6eq 1560	1 |- (F Fn (A X. B) -> ran F = {z | E.x e. A E.y e. B z = (xFy)})

Syntax hints:   -> wi 3   = wceq 988  {cab 1499  E.wrex 1684  <.cop 2456   X. cxp 3223  ran crn 3226   Fn wfn 3232  ` cfv 3237  (class class class)co 4039

This theorem is referenced by:  oprvalelrn 4117

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-sep 2754  ax-pow 2794  ax-pr 2832

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-ral 1687  df-rex 1688  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-uni 2552  df-br 2670  df-opab 2718  df-id 2889  df-xp 3239  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-res 3245  df-ima 3246  df-fun 3247  df-fn 3248  df-fv 3253  df-opr 4041

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